Abstract

Let be a nonempty subset of a Hausdorff topological vector space , and let be a real-valued continuous function on . If for each , there exists such that , then is called -simultaneously proximal and is called -best simultaneous approximation for in . In this paper, we study the problem of -simultaneous approximation for a vector subspace in . Some other results regarding -simultaneous approximation in quotient space are presented.

1. Introduction

Let be a closed subset of a Hausdorff topological vector space and a real-valued continuous function on . For , set . A point is called -best approximation to in if . The set denotes the set of all -best approximations to in . Note that this set may be empty. The set is said to be -proximal (-Chebyshev) if for each , is nonempty (singleton). The notion of -best approximation in a vector space was given by Breckner and Brosowski [1] and in a Hausdorff topological space by Narang [2, 3]. For a Hausdorff locally convex topological vector space and a continuous sublinear functional on , certain results on best approximation relative to the functional were proved in [1, 4]. By using the existence of elements of -best approximation, certain results on fixed points were proved by Pai and Veermani in [5]. In addition, for a topological vector space relative to upper semicontinuous functions, some results on best approximation were proved by Haddadi and Hamzenejad [6]. Moreover, Naidu [7] proved some results on best simultaneous approximation related to -nearest point and topological vector space .

Analogous to the problem of simultaneous approximation [8], we introduce the concept of best -simultaneous approximation as follows.

Definition 1. Let be a non-empty subset of a Hausdorff topological vector space , and let be a real-valued continuous function on . A point is called -best simultaneous approximation in if there exists such that

The set of all -best simultaneous approximations to in is denoted by

The set is called -simultaneously proximal (-simultaneously Chebyshev) if for each , (singleton). If , simultaneous -proximal is precisely -proximal.

We remark that if , then the concept of -best approximation is precisely the best approximation.

A set is said to be inf-compact at a point [5] if each minimizing sequence in (i.e., has a convergent subsequence in . The set is called inf-compact if it is inf-compact at each .

It is easy to see that if is compact or inf-compact, then is -simultaneously proximal.

In this paper, we introduce the concept of -simultaneous approximation and study the existence and uniqueness problem of -simultaneous approximation of a subspace of a Hausdorff topological vector space . Certain results regarding -simultaneous approximation in quotient spaces are obtained by generalizing some of the results in [9].

Throughout this paper, is a Hausdorff topological vector space and is a real-valued continuous function on .

2. -Simultaneous Approximation

In this section, we give some characterizations of -proximal sets in . We begin with the following definitions.

Definition 2. A function is called absolutely homogeneous if , for all and all .

Definition 3. A subset of is called -closed if for all sequences of and for all , such that , we have .

Definition 4. A subset of is called -compact if for every sequence in there exist a subsequence of and such that .

Definition 5. For , where and , is said to be -orthogonal to denoted by , if for every scalar . Also, is said to be -orthogonal to a set if , for all .

Definition 6. We say that is -compact if every net in has a convergent subnet.

Theorem 7. Let be a subset of . Then, one has the following.(1), for all , where .(2), for all .(3) is -simultaneously proximal -simultaneously Chebyshev if and only if is -simultaneously proximal -simultaneously Chebyshev for every .Moreover, if is absolutely homogeneous function, then one has the following.(4), for all and .(5), for all and .(6) is -simultaneously proximal -simultaneously Chebyshev) if and only if is -simultaneously proximal (-simultaneously Chebyshev), .(7) If is convex function and is a convex set, then is convex.

Proof. Let and . Then
The equation
implies that if and only if . Thus,
The proof follows immediately from part above. Let , . Then,
If , then we are done. If and , then and
This implies that
which implies that . The proof follows immediately from part above. Let , . Since is convex, then . We must show that ; that is,
So,
which implies that is convex.

Example 8. Let and , and let . If , then one can show that .

Theorem 9. Let be an absolutely homogeneous real-valued function on and a vector subspace of . Then,(1), for all , ;(2), for all , .

Proof. Let . Then,
Let . Then,
if and only if
for all , which implies that , so, .

Theorem 10. Let be a positive real-valued function on such that if and only if . Then, if is -simultaneously proximal, then is -closed.

Proof . Since is a positive function, then for all . Let be a sequence of and , such that . This implies that
Since is -simultaneously proximal, then there exists such that
Hence, for all, .Using the assumption it follows that , and, hence, . Consequently, and is -closed.

Theorem 11. Let be a topological vector space and a vector subspace of . Suppose that is continuous function and is -compact; then, is -simultaneously proximal.

Proof. Let . Since
then, for any constant , there exists such that
But is -compact; then, there exists a subnet such that . Thus,
Since is continuous, then
Also,
Hence, .

For a subset of , let us define to be such that

Example 12. Consider and . Let ; then, one can see that

Using the previous definition of , we prove the following theorem characterizing -simultaneously proximal subspaces.

Theorem 13. Let be a vector subspace of . Then, is -simultaneously proximal in if and only if , where .

Proof. Suppose that . Then, for , there exists and such that . Hence, , and
and so
So, is -simultaneously proximal.Conversely, suppose that is -simultaneously proximal and . Then, there exists such that
where . If , then
which implies that and .

Proposition 14. Let be a topological vector space and -simultaneous proximal subset of . Then,(1) if and only if ;(2)if is symmetric (i.e., for all ), then if and only if ;(3)if , then , where ;(4)if and , then , where .

Proof. Let if and only if .Thus,
which implies that . Let . Since is symmetric, then
Hence, , which implies that
Let . Since , then
So,
Hence, . Let and . Then,
Thus,
Hence, .

Theorem 15. Let be a vector subspace of . If , then is -simultaneously proximal, where is the canonical map .

Proof. Let and . Then, for some . Hence, for some . Thus, . Therefore, . By Theorem 15, is -simultaneously proximal.

3. -Simultaneous Approximation in Quotient Space

Definition 16. Let and be two vector subspaces of such that is closed and . Suppose that is a positive real-valued function defined on . Then, a function can be defined as follows:
for each .

Theorem 17. Let and be two vector subspaces of such that . If is a point of -best simultaneous approximation to in , then is an -best simultaneous approximation to in.

Proof. Suppose that is not -best simultaneous approximation to in . Then,
for at least , say , such that
Since
we have
Thus, for some , we have
so,
Since implies that , therefore, is not -best simultaneous approximation to in , which is a contradiction.

Corollary 18. Let and be two vector subspaces of such that . Then, if is -simultaneously proximal in , then is -simultaneously proximal in .

Proof. If is -simultaneously proximal in , then there exists at least such that is -best simultaneous approximation to in . Thus by Theorem 11, is an -best simultaneous approximation to in , so, is -simultaneously proximal in .

Theorem 19. Let and be two vector subspaces of such that . If is -simultaneously proximal in and is -simultaneously proximal in , then is -simultaneously proximal in .

Proof. Since is -simultaneously proximal in , then there exists such that is -best simultaneous approximation to from , so,
for all . Note that
Since is -simultaneously proximal in , then there exists such that
for all and . So,
for all and . Hence,
So, is an -best simultaneous approximation to from and is -simultaneously proximal in .

Theorem 20. Let and be two vector subspaces of such that . If is -simultaneously proximal in and is -simultaneously Chebyshev in , then is -simultaneously Chebyshev in .

Proof. Suppose not, then there exists , and , such that . Thus, . Since is -simultaneously proximal in , then
Let and . By Theorem 13, and are -best simultaneous approximation to from . Since is -simultaneously Chebyshev in , then , and, hence, , which is a contradiction.

Theorem 21. Let and be two vector subspaces of a topological vector space . If is -simultaneously Chebyshev in , then the following assertions are equivalent:(i) is -simultaneously Chebyshev in ;(ii) is simultaneously Chebyshev in .

Proof. () By hypothesis, is -simultaneous Chebyshev. Assume that is not -simultaneous Chebyshev in . Then, there exists which has two distinct -best simultaneous approximations, say and . Thus, we have and . Since , we have that and . By hypothesis, is -simultaneous Chebyshev, and so . Then, there exists such that . Thus, we conclude that
So, and are -best simultaneous approximations to from . Hence, is not -simultaneously Chebyshev. This is a contradiction.() Assume that does not hold. Then, there exists which has two distinct -best simultaneous approximations, say and ; thus, . Since is -simultaneously proximal, so there exist -best simultaneous approximations and to and from , respectively. Therefore, we have and . Since , and , so and . But is -simultaneously Chebyshev. Thus we get , and therefore . This is a contradiction.

Definition 22. A subset of is called -quasisimultaneously Chebyshev if is non-empty and -compact set in , for all .

Theorem 23. Let be a positive function, an -simultaneously proximal vector subspace of , and K -quasisimultaneously Chebyshev of such that . Then, is -quasi-simultaneously Chebyshev in .

Proof. Since is -simultaneously proximal in , then by Corollary 12, is -simultaneously proximal in . Let and a sequence in . For every , there exists such that . But since is a vector subspace, we have
Since is -quasi-simultaneously Chebyshev of , the sequence has a subsequence which is -convergent to , meaning that
But
Hence,
Consequently, is -compact and is -quasi-simultaneously Chebyshev. This completes the proof.

Acknowledgments

The authors would like to express their appreciation and gratitude to the editor and the anonymous referees for their comments and suggestions for this paper. Also, the authors wish to thank Professor Sharifa Al-Sharif for her valuable suggestions and comments.